To graph the function [tex]\( f(x) = \sqrt[3]{x + 2} + 4 \)[/tex], we will follow these steps:
1. Understand the function:
- The expression [tex]\(\sqrt[3]{x + 2}\)[/tex] represents the cube root of [tex]\(x + 2\)[/tex].
- The function is then shifted vertically upward by 4 units due to the "+ 4".
2. Identify key features:
- Domain: The cube root function [tex]\(\sqrt[3]{x + 2}\)[/tex] is defined for all real [tex]\(x\)[/tex], so the domain of [tex]\(f(x)\)[/tex] is [tex]\( (-\infty, \infty) \)[/tex].
- Range: Since a cube root can take any real number, adding 4 to it will shift all outputs up by 4. Thus, the range is also [tex]\((-\infty, \infty)\)[/tex].
- Intercepts:
- Y-intercept: Plug [tex]\( x = 0 \)[/tex] into the function: [tex]\( f(0) = \sqrt[3]{0 + 2} + 4 = \sqrt[3]{2} + 4 \)[/tex].
- X-intercept: Set [tex]\( f(x) = 0 \)[/tex] and solve for [tex]\( x \)[/tex]: [tex]\(\sqrt[3]{x + 2} + 4 = 0\)[/tex], which simplifies to [tex]\(\sqrt[3]{x + 2} = -4\)[/tex]. Cubing both sides gives [tex]\( x + 2 = -64 \)[/tex], so [tex]\( x = -66 \)[/tex].
3. Plot points:
- Evaluate the function at various points to get an idea of the shape of the graph.
- Example points:
- When [tex]\( x = -10 \)[/tex]: [tex]\( f(-10) = \sqrt[3]{-10 + 2} + 4 = \sqrt[3]{-8} + 4 = -2 + 4 = 2 \)[/tex].
- When [tex]\( x = 0 \)[/tex]: [tex]\( f(0) = \sqrt[3]{2} + 4 \approx 4.26 \)[/tex].
- When [tex]\( x = 10 \)[/tex]: [tex]\( f(10) = \sqrt[3]{10 + 2} + 4 = \sqrt[3]{12} + 4 \approx 2.29 + 4 = 6.29 \)[/tex].
4. Graph the function:
- Draw the x-axis and y-axis.
- Plot the intercept points found, such as the y-intercept at [tex]\((0, \sqrt[3]{2} + 4)\)[/tex].
- Plot additional points calculated (e.g., [tex]\((-10, 2)\)[/tex], [tex]\( (10, 6.29) \)[/tex]).
- Smoothly connect these points, keeping in mind the typical shape of the cube root function, which tends to increase or decrease gently and is symmetric about the origin if not shifted.
Here is an illustration:
```
f(x)
|
10|
|
8|
|
6|
|
4|------------------------
|
2| *
|
0|____________________________________ x
-66 -10 0 10
```
- X-intercept: approximately at [tex]\( (-66, 0) \)[/tex]
- Y-intercept: approximately at [tex]\( (0, \sqrt[3]{2} + 4) \)[/tex]
The curve starts from bottom-left, passing through [tex]\(( -66, 0 )\)[/tex], then gradually rising towards the top-right, approximating the cube root function shape but shifted up by 4 units.
By following these steps, you can create a detailed and accurate graph of the function [tex]\( f(x) = \sqrt[3]{x + 2} + 4 \)[/tex].
